On Mori chamber and stable base locus decompositions
aa r X i v : . [ m a t h . AG ] S e p ON MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS
ANTONIO LAFACE, ALEX MASSARENTI, AND RICK RISCHTER
Abstract.
The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Morichamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter.We investigate, from both the geometrical and the combinatorial viewpoints, the differences between thesedecompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide fora Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picardrank two for which the decompositions are different, showing that our criterion is sharp. Finally, we classify thesmooth toric -folds of Picard rank three for which the two decompositions are different. Contents
1. Introduction 12. Mori chamber and stable base locus decompositions 43. Examples 84. Smooth toric 3-folds of Picard rank three 145. Mori dream spaces of Picard rank two 166. Grassmannians blow-ups 22References 251.
Introduction
Mori dream spaces , introduced by Y. Hu and S. Keel in [HK00], are varieties whose total coordinate ring,called the
Cox ring , is finitely generated. The birational geometry of a Mori dream space is encoded in its coneof effective divisors together with a chamber decomposition on it, called
Mori chamber decomposition . Twoeffective divisors lie in the interior of the same Mori chamber if there is an isomorphism between the targetspaces of the corresponding dominant rational maps making the obvious triangular diagram commutative.The birational geometry of a Mori dream spaces can also be described via the Variation of Geometric InvariantTheory of its Cox ring. As proven in [HK00], [ADHL15, Section 3.3.4], and [HKP06, Appendix A] in the caseof complete toric varieties using the volume function, from this point of view GIT chambers correspond to Morichambers.The pseudo-effective cone of a projective variety with zero irregularity, so in particular of a Mori dream space,can be decomposed into chambers depending on the stable base locus of the corresponding linear series. Suchdecomposition, called stable base locus decomposition , in general is coarser than the Mori chamber decomposition.The Mori theory of important classes of moduli spaces such as moduli of curves [Has05], [HH09], [HH13],Hilbert schemes of points on surfaces [BC13], [ABCH13], Kontsevich spaces of stable maps [Che08], [CC10],[CC11], spaces of complete forms [Hue15], [Mas18a], [Mas18b], and moduli spaces of parabolic bundles [Muk05],[AM16] have recently been studied in a series of papers.In this paper, given a Mori dream space X , we aim to understand how far is the stable base locus decompo-sition of Eff p X q from determining its Mori chamber decomposition. In Section 3 we produce examples of Moridream spaces for which the two decompositions are different and we interpret them both from the geometricand the combinatorial viewpoints. Date : September 13, 2019.2010
Mathematics Subject Classification.
Primary 14E05, 14L10, 14M15; Secondary 14J45, 14MXX.
Key words and phrases.
Mori dream spaces, Mori chamber decomposition, stable base locus decomposition, Cox rings.
While producing examples of either non compact varieties or of varieties with Picard rank greater than orequal to three turns out to be fairly feasible, it is quite tricky to exhibit a normal Q -factorial projective Moridream space of Picard rank two for which the Mori chamber decomposition is a non trivial refinement of thestable base locus decomposition. In Example 3.6 we construct such a Mori dream space, and at the best of ourknowledge this is the fist example of a projective variety displaying this particular behavior appearing in theliterature. Theorem 1.1.
Let Z be the toric variety with Cox ring K r T , . . . , T s whose grading matrix and irrelevantideal are the following Q “ „ J irr p Z q “ x T , T y X x T , . . . , T y and let F, G be two general polynomials of degree p , q in the T i . Then the ring K r T , . . . , T sp F, G q is the Cox ring of a projective normal Q -factorial Mori dream space X Ă Z of Picard rank two. Furthermore,the Mori chamber decomposition of Eff p X q consists of three chambers while its stable base locus decompositionconsists of two chambers. By Proposition 2.13 if X is a toric -fold such that the two decompositions differ inside the movable conethen there are at least five Mori chambers in the movable cone, and Examples 3.8, 3.9 show that Proposition2.13 is sharp meaning that one can have that the two decompositions differ inside the movable cone with fivechambers and with three chambers in dimension higher than three. In Section 4 we restrict to the smooth case,and we classify smooth toric -folds of Picard rank three such that their Mori chamber and stable base locusdecomposition do not coincide. Theorem 1.2.
Let X be a smooth toric -fold of Picard rank three such that its Mori chamber and stable baselocus decomposition do not coincide. Then the Mori chamber decomposition of X is one of the seven types listedin the following table. grading matrix Effective cone grading matrix Effective cone G : “ »– α β fifl α, β ą G : “ »– α β γ fifl α, β, γ ă G : “ »– γ ´ fifl γ ą G : “ »– β γ ´ fifl β ą ą γG : “ »– β γ ´ fifl β ă ă γ G : “ »– β γ ´ fifl β ă γ ´ ă ´ N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 3 G : “ »– ´ fifl Where, in the pictures, the black region is the semi-ample cone and the two gray regions are the GIT chamberssharing the same stable base locus. In particular, for any smooth toric -fold the Mori chamber and the stablebase locus decomposition coincide inside the movable cone. In Section 5 we focus on Mori dream spaces of Picard rank two. Recall that a Mori dream space can berecovered as a GIT quotient, with respect to a suitable polarization, of the spectrum of its Cox ring by a torus.In the fist part of Section 5, assuming that the Picard rank is two, we reach a simple description of the nonsemi-stable loci with respect to all the possible polarizations, and of the stable base loci of effective divisorsin terms of the generators of the Cox ring. Thanks to these characterizations in Theorem 5.8 we get technicalcriteria on the non semi-stable loci aimed to establish whether the Mori chamber and the stable base locusdecomposition of a given Mori dream space of Picard rank two coincide. As observed in Corollary 5.13 theirreducibility of the non semi-stable loci is a sufficient condition for the two decompositions to coincide.In Theorem 5.14 we prove that under suitable inequalities, that need just the knowledge of the generators ofthe Cox ring in order to be checked, the two decompositions coincide. Furthermore, in Proposition 5.2 we getanother criterion for the equality of the decompositions. The usefulness of these results lies in the fact that ingeneral, even in Picard rank two, the stable base locus decomposition is considerably easier to compute thanthe Mori chamber decomposition.Note that if X is a projective Mori dream space of Picard rank two we can fix a total order on the classes inthe effective cone: w ď w if w is on the left of w . Given two convex cones λ, λ contained in the effective conewe will write λ ď λ if w ď w for any w P λ and w P λ . Denote by t f , . . . , f r u a minimal set of homogeneousgenerators for the Cox ring R p X q of X , and let w i “ deg p f i q for any i .The criteria in the Proposition 5.2, Theorem 5.14 and Corollary 5.15 can be summarized in the followingstatement. Theorem 1.3.
Let X be a Q -factorial Mori dream space with Picard rank two, t f , . . . , f r u a minimal set ofhomogeneous generators for the Cox ring R p X q , w i : “ deg p f i q , and λ A be the ample chamber of X . Denote by c the codimension of X into its canonical toric embedding [ADHL15, Section 3.2.5] . Define h ` : “ t f i : w i ě λ A u and h ´ : “ t f i : w i ď λ A u If one of the following two conditions is satisfied(i) all the generators of R p X q appear in the walls of the stable base locus decomposition of Eff p X q ,(ii) h ´ ą c and h ` ą c ,then the Mori chamber decomposition and the stable base locus decomposition of Eff p X q coincide.In particular, if Z is a projective normal Q -factorial toric variety with rk p Cl p Z qq “ , and X Ď Z is aprojective normal Q -factorial Mori dream hypersurface such that ı ˚ : Cl p Z q Ñ Cl p X q is an isomorphism, thenthe Mori chamber and the stable base locus decompositions of both Eff p Z q and Eff p X q coincide. As observed in Remark 5.16, Theorem 1.1 shows that the bounds in Theorem 1.3 item p ii q can not beimproved. Indeed in Theorem 1.1 we have h ` “ c “ .In Subsection 5.17 we show how Theorem 1.3 item p ii q , together with the classification of Picard rank twovarieties, with a torus action of complexity one in [FHN16, Theorem 1.1] immediately implies that the Morichamber decomposition is equal to the stable base locus decomposition for this class of varieties. Note that, asshown by the examples in Subsection 5.17, Lemma 5.5 along with the proof of Theorem 5.8 provide a concretemethod to compute Mori chamber decompositions.We would like so stress that these results can also be useful in order to compute the Sarkisov factorizationof a birational map X Y between two Q -factorial Fano varieties of Picard rank one. Indeed, if there existsa Mori dream space Z of Picard rank two admitting a dominant morphism Z Ñ X then such a factorization is ANTONIO LAFACE, ALEX MASSARENTI, AND RICK RISCHTER determined by a so called -ray game on Z , and such a -ray game is in turn determined by the Mori chamberdecomposition of Eff p Z q . We refer to [Cor95], [HM13], [AZ16], [Ahm17] for details on this topic and explicitexamples.Finally, in Section 6 we apply Theorem 1.3 item p i q to show that the Mori chamber decomposition of the blow-up G p r, n q of the Grassmannian G p r, n q , parametrizing r -planes in P n , at a point coincides with its stable baselocus decomposition, and can be described in terms of linear systems of hyperplanes containing the osculatingspaces of G p r, n q at the blown-up point. This provides a positive answer to [MR18, Question 6.9].All through the paper we will work over an algebraically closed field K of characteristic zero, and givena Q -factorial Mori dream space X we will denote by MCD p X q and SBLD p X q respectively the Mori chamberdecomposition and the stable base locus decomposition of its effective cone. Acknowledgments.
We thank the referee for giving us suggestions that lead us to Section 4, Proposition 2.12,Proposition 2.13 and Examples 3.8 and 3.9.The first named author was partially supported by Proyecto FONDECYT Regular N. 1150732, ProyectoFONDECYT Regular N. 1190777 and by project Anillo ACT 1415 PIA Conicyt. The second named author is amember of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the IstitutoNazionale di Alta Matematica "F. Severi" (GNSAGA-INDAM).2.
Mori chamber and stable base locus decompositions
Let X be a normal projective variety over an algebraically closed field of characteristic zero. We denote by N p X q the real vector space of R -Cartier divisors modulo numerical equivalence. The nef cone of X is theclosed convex cone Nef p X q Ă N p X q generated by classes of nef divisors. The movable cone of X is the convexcone Mov p X q Ă N p X q generated by classes of movable divisors . These are Cartier divisors whose stable baselocus has codimension at least two in X . The effective cone of X is the convex cone Eff p X q Ă N p X q generatedby classes of effective divisors . We have inclusions Nef p X q Ă Mov p X q Ă Eff p X q .We will denote by N p X q be the real vector space of numerical equivalence classes of -cycles on X . Theclosure of the cone in N p X q generated by the classes of irreducible curves in X is called is called the Mori cone of X , we will denote it by NE p X q .A class r C s P N p X q is called moving if the curves in X of class r C s cover a dense open subset of X . Theclosure of the cone in N p X q generated by classes of moving curves in X is called the moving cone of X andwe will denote it by mov p X q . We refer to [Deb01, Chapter 1] for a comprehensive treatment of these topics.We say that a birational map f : X X to a normal projective variety X is a birational contraction if itsinverse does not contract any divisor. We say that it is a small Q -factorial modification if X is Q -factorial and f is an isomorphism in codimension one. If f : X X is a small Q -factorial modification, then the naturalpull-back map f ˚ : N p X q Ñ N p X q sends Mov p X q and Eff p X q isomorphically onto Mov p X q and Eff p X q ,respectively. In particular, we have f ˚ p Nef p X qq Ă Mov p X q . Definition 2.1.
A normal projective Q -factorial variety X is called a Mori dream space if the following condi-tions hold:-
Pic p X q is finitely generated, or equivalently h p X, O X q “ ,- Nef p X q is generated by the classes of finitely many semi-ample divisors,- there is a finite collection of small Q -factorial modifications f i : X X i , such that each X i satisfiesthe second condition above, and Mov p X q “ Ť i f ˚ i p Nef p X i qq .By [BCHM10, Corollary 1.3.2] smooth Fano varieties are Mori dream spaces. In fact, there is a larger classof varieties called log Fano varieties which are Mori dream spaces as well. By the work of M. Brion [Bri93] wehave that Q -factorial spherical varieties are Mori dream spaces. An alternative proof of this result can be foundin [Per14, Section 4].The collection of all faces of all cones f ˚ i p Nef p X i qq in Definition 2.1 forms a fan which is supported on Mov p X q . If two maximal cones of this fan, say f ˚ i p Nef p X i qq and f ˚ j p Nef p X j qq , meet along a facet, then thereexist a normal projective variety Y , a small modification ϕ : X i X j , and h i : X i Ñ Y and h j : X j Ñ Y small birational morphisms of relative Picard number one such that h j ˝ ϕ “ h i . The fan structure on Mov p X q can be extended to a fan supported on Eff p X q as follows. Definition 2.2.
Let X be a Mori dream space. We describe a fan structure on the effective cone Eff p X q , calledthe Mori chamber decomposition . We refer to [HK00, Proposition 1.11] and [Oka16, Section 2.2] for details.
N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 5
There are finitely many birational contractions from X to Mori dream spaces, denoted by g i : X Y i . Theset Exc p g i q of exceptional prime divisors of g i has cardinality ρ p X { Y i q “ ρ p X q ´ ρ p Y i q . The maximal cones C of the Mori chamber decomposition of Eff p X q are of the form: C i “ @ g ˚ i ` Nef p Y i q ˘ , Exc p g i q D . We call C i or itsinterior C ˝ i a maximal chamber of Eff p X q . Definition 2.3.
Let X be a normal projective variety with finitely generated divisor class group Cl p X q : “ WDiv p X q{ PDiv p X q , in particular h p X, O X q “ . The Cox sheaf and
Cox ring of X are defined as R : “ à r D sP Cl p X q O X p D q R p X q : “ Γ p X, R q Recall that R is a sheaf of Cl p X q -graded O X -algebras, whose multiplication maps are discussed in [ADHL15,Section 1.4]. In case the divisor class group is torsion-free one can just take the direct sum over a subgroup of WDiv p X q , isomorphic to Cl p X q via the quotient map, getting immediately a sheaf of O X -algebras. Denote by p X the relative spectrum of R and by X the spectrum of R p X q . The Cl p X q -grading induces an action of thequasi-torus H X : “ Spec C r Cl p X qs on both spaces. The inclusion O X Ñ R induces a good quotient p X : p X Ñ X with respect to this action. Summarizing we have the following diagram p X Ď XX p X to which we will refer as the Cox construction of X . In case R p X q is a finitely generated algebra the complementof p X in the affine variety X has codimension ě . This subvariety is the irrelevant locus and its defining idealis the irrelevant ideal J irr p X q Ď R p X q . Remark 2.4.
By [HK00, Proposition 2.9] a normal and Q -factorial projective variety X over an algebraicallyclosed field K , with finitely generated Picard group is a Mori dream space if and only if R p X q is a finitelygenerated K -algebra. Furthermore, the following equality holds dim R p X q “ dim p X q ` rank Cl p X q see for instance [ADHL15, Theorem 3.2.1.4].Let X be a normal Q -factorial projective variety, and let D be an effective Q -divisor on X . The stable baselocus B p D q of D is the set-theoretic intersection of the base loci of the complete linear systems | sD | for allpositive integers s such that sD is integral B p D q “ č s ą B p sD q . Since stable base loci do not behave well with respect to numerical equivalence, we will assume that h p X, O X q “ so that linear and numerical equivalence of Q -divisors coincide.Then numerically equivalent Q -divisors on X have the same stable base locus, and the pseudo-effective cone Eff p X q of X can be decomposed into chambers depending on the stable base locus of the corresponding linearseries called stable base locus decomposition , see [CdFG17, Section 4.1.3] for further details.If X is a Mori dream space, satisfying then the condition h p X, O X q “ , determining the stable base locusdecomposition of Eff p X q is a first step in order to compute its Mori chamber decomposition. Remark 2.5.
Recall that two divisors D , D are said to be Mori equivalent if B p D q “ B p D q and thefollowing diagram of rational maps is commutative XX p D q X p D q Ă φ D φ D where the horizontal arrow is an isomorphism. Therefore, the Mori chamber decomposition is a refinement ofthe stable base locus decomposition. ANTONIO LAFACE, ALEX MASSARENTI, AND RICK RISCHTER
Let X be a Mori dream space with Cox ring R p X q and grading matrix Q . The matrix Q defines a surjection Q : E Ñ Cl p X q from a free, finitely generated, abelian group E to the divisor class group of X . Denote by γ the positivequadrant of E Q : “ E b Z Q . Let e , . . . , e r be the canonical basis of E Q . Given a face γ ĺ γ we say that i P t , . . . , r u is a cone index of γ if e i P γ . The face γ is an F -face if there exists a point of X “ Spec p R p X qq whose i -th coordinate is non-zero exactly when i is a cone index of γ [ADHL15, Construction 3.3.1.1]. The setof these points is denoted by X p γ q . Example 2.6. If R p X q “ K r T ,...,T sx T T ` T ` T T y then γ “ cone p e , e q is an F -face and X p γ q “ tp x , , , x , q P X : x x ‰ u On the other hand, cone p e , e , e q is not an F -face.Given the Cox construction of X we denote by X p γ q Ď X the image of X p γ q , and given an F -face γ itsimage Q p γ q Ď Cl p X q Q is an orbit cone of X . The set of all orbit cones of X is denoted by Ω . Accordingly to[ADHL15, Definition 3.1.2.6] a class w P Cl p X q defines the GIT chamber (2.7) λ p w q : “ č t ω P Ω : w P ω u ω If w is an ample class of X the corresponding GIT chamber is the semi-ample cone of X . The variety X canbe reconstructed from the pair p R p X q , Φ q formed by the Cox ring together with a bunch of cones , consisting ofcertain subsets of the orbit cones [ADHL15, Definition 3.1.3.2]. According to [ADHL15, Example 3.1.3.6] everyGIT chamber λ defines a bunch of orbit cones Φ p λ q : “ t ω P Ω : ω ˝ Ě λ ˝ u Given a class w P Cl p X q we denote by λ sbl p w q the subset of Cl p X q Q consisting of all classes which have thesame stable base locus of w . Proposition 2.8.
Let X be a normal variety with finitely generated Cox ring, bunch of orbit cones Φ , and let w P Cl p X q be a class of X . Then λ sbl p w q “ č t ω P Φ : w P ω u ω X č t ω P Φ : w R ω u ω c Proof.
Recall that, according to [ADHL15, Construction 3.2.1.3], the set of relevant faces rlv p Φ q is the set offaces of γ which are mapped by Q to elements of Φ . Each relevant face γ ĺ γ defines a subset X p γ q Ď X consisting of all the points of X whose i -th Cox coordinate is non-zero exactly when i is a cone index of γ [ADHL15, Construction 3.3.1.1]. By [ADHL15, Proposition 3.3.2.8] the stable base locus of a class w is theunion(2.9) B p w q : “ ď t γ P rlv p Φ q : w R Q p γ qu X p γ q Applying Q to the elements of the set t γ P rlv p Φ q : w R Q p γ qu one gets the set t ω P Φ : w R ω u and theformer set is completely determined by the latter. We claim that two classes w, w define the same stable baselocus if and only if the following holds(2.10) t ω P Φ : w P ω u “ t ω P Φ : w P ω u Clearly, if w, w define the same stable base locus then (2.10) holds. Now, assume that there is an ω P Φ such that w R ω and w P ω . Let γ ĺ γ be such that Q p γ q “ ω . It suffices to show that X p γ q is notcontained in the stable base locus of w . Indeed if X p γ q is any strata which contains X p γ q then γ ĺ γ , sothat w P ω “ Q p γ q Ď Q p γ q . Thus X p γ q does not appear in (2.9), and the claim is proved. Finally, thestatement follows by observing that if w P č t ω P Φ : w P ω u ω X č t ω P Φ : w R ω u ω c then the cones which contain, respectively do not contain w are the same of those which contain, respectivelydo not contain w . (cid:3) N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 7
Corollary 2.11.
Let X be a normal variety with finitely generated Cox ring, bunch of orbit cones Φ and let w P Cl p X q be a class of X . Then the following inclusion holds λ p w q Ď λ sbl p w q Proof.
It is a direct consequence of Proposition 2.8 and of the following equalities λ p w q “ č t ω P Ω : w P ω u ω “ č t ω P Ω : w P ω u ω X č t ω P Ω : w R ω u ω c where the first is by definition while the second is due to the fact that any two classes in the relative interior λ p w q determine the same chamber. (cid:3) Proposition 2.12.
Let X be a normal variety with finitely generated Cox ring, bunch of orbit cones Φ , and w , w P Cl p X q . Then the following are equivalent:(1) λ sbl p w q “ λ sbl p w q ;(2) t ω P Φ : w P ω u “ t ω P Φ : w P ω u ;(3) Ş t ω P Φ : w P ω u ω “ Ş t ω P Φ : w P ω u ω .Moreover, if Φ “ Φ p λ q , with λ distinct from λ and λ , then each of the above condition is implied by cone p λ Y λ q X ˚ λ ‰ and cone p λ Y λ q X ˚ λ ‰ .Proof. The equivalence of p q with p q is given by the proof of Proposition 2.8, and the implication p q ñ p q isclear. To prove p q ñ p q it suffices to observe that if there is an ω P Ω such that w P ω but w R ω , then w would be contained in Ş t ω P Φ : w P ω u ω but not in Ş t ω P Φ : w P ω u ω , so that the two sets will be different. Finally if cone p λ Y λ q X ˚ λ ‰ holds then any orbit cone ω P Φ p λ q which contains λ must intersect the interior of λ sothat λ Ď ω . Thus t ω P Φ : w P ω u Ď t ω P Φ : w P ω u holds. Similarly one proves the opposite inclusion. (cid:3) Proposition 2.13.
Let X be a toric -fold such there are two Mori chambers in Mov p X q whose union gives asingle stable base locus chamber. Then there are at least five Mori chambers inside Mov p X q .Proof. Assume there are four Mori chambers inside
Mov p X q , and consider the chamber corresponding to Nef p X q .If the three remaining chambers C , C , C are adjacent to Nef p X q , and C , C , C are the corresponding threedistinct flipping curves, then by Nakamaye’s theorem [Laz04, Theorem 10.3.5] we get that the stable base locusof a divisor in C i is C i , and hence the stable base loci of divisors in C , C , C are distinct. Note that thisargument works also for three or less chambers.If C , C are adjacent to Nef p X q and C is not then again Nakamaye’s theorem yields that divisors in C , C have as base loci two different irreducible curves. Furthermore, the base locus of a divisor in C is a curve withtwo components.Now, let C , . . . , C be four consecutive maximal Mori chambers contained in the moving cone of a Q -factorialtoric threefold X of Picard rank n and let X “ X , . . . , X be the corresponding birational models of X . Assumethat C is the ample cone of X . Let τ ij : “ C i X C j and let Γ ij Ď X j be the new irreducible curve produced bythe wall crossing of τ ij . By the previous argument the two chambers with the same stable locus are C and C .In order for C and C to have the same stable base locus on X the center Γ Ď X of the small Q -factorialmodification given by the wall crossing of τ must be an irreducible curve which does not exists in X . Thisis possible only if this curve is the strict transform of the curve Γ created by the wall crossing of τ . Inparticular Γ „ α Γ ` β Γ , with α, β P Q . If we denote by H ij the linear span of the cone τ ij in Cl Q p X q , then the above discussion impliesthat the three hyperplanes H , H , H lie on a pencil. Each of these hyperplanes must contain at least n ´ of the n ` classes of generators of the Cox ring. Assuming that the intersection H X H X H contains r of these classes then we have the following inequality p n ´ ´ r q ` r ď n ` which implies r ě n ´ . If r “ n ´ then each of the three hyperplanes must contain exactly two more classesof generators. Up to symmetries the possible configurations of the six classes on the three hyperplanes are thefollowing. ANTONIO LAFACE, ALEX MASSARENTI, AND RICK RISCHTER
In the last four cases at least one of the facets τ ij would be not contained in the interior of the moving cone (seeProposition [ADHL15, 3.3.2.3]). In the first two cases there would be at least six Mori chambers in the movingcone.If r “ n ´ then each of the three hyperplanes contains at least one more class. The possible configurationsare the following, where in each case still two more generators have to be added to the picture.In the first case it is possible to fulfill these conditions only if the two more classes are as in the picture below.In this case the moving cone contains at least five Mori chambers, as shown in Example 3.8. In the last case, upto symmetry there is again one possibility for the open chambers where the two new classes can belong. Thisis displayed in the picture below.However, in this last case we can have either five or three chambers in the movable cone.If r “ n ´ then again each of the three hyperplanes contains at least one more class, so that the aboveconfigurations still apply. Here we have to add only one more class to the picture and thus one of the threecones τ ij would not be contained in the interior of the moving cone. (cid:3) Examples 3.8 and 3.9 in the next section show that Proposition 2.13 is sharp in the sense that one can hasMori chamber decomposition distinct from the stable base locus decomposition inside the movable cone withfive chambers and with three chambers in dimension higher than three.3.
Examples
In this section we give examples of varieties for which the Mori chamber and the stable base locus decompo-sition do not coincide, and we analyze this phenomenon from both the geometrical and the combinatorial pointof view.
Example 3.1. (Birational viewpoint) Consider a plane Π Ă P n and five general points p , . . . , p P Π . Let f : X Ñ P n be the blow-up of P n at p , . . . , p with exceptional divisors E , . . . , E . Then the strict transform r Π Ă X of Π is a del Pezzo surface of degree four. In particular r Π is a Mori dream space.Let e , . . . , e be classes of a line in the exceptional divisors, and l the pull-back a a general line in P n . Let r C Ă X be an irreducible curve. If r C gets contracted by f the r C „ me i with m ą for some i P t , . . . , u .Otherwise, we may write r C „ dl ´ m e ´ ¨ ¨ ¨ ´ m e , that is r C is the strict transform of a curve C Ă P n ofdegree d having multiplicity m i at p i for i “ , . . . , .If d ă m ` ¨ ¨ ¨ ` m then C Ă Π and r C Ă r Π . In this case we may write r C as a linear combination withnon-negative coefficients of e , . . . , e , l ´ e i ´ e j , l ´ e ´ ¨ ¨ ¨ ´ e since these are the generators of NE p r Π q . If d ě m ` ¨ ¨ ¨ ` m then we may write r C „ m p l ´ e q ` . . . m p l ´ e q ` p d ´ m ´ ¨ ¨ ¨ ´ m q l where l ´ e i “ p l ´ e i ´ e j q ` e j . Therefore, NE p X q “ x e i , l ´ e i ´ e j , l ´ e ´ ¨ ¨ ¨ ´ e y N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 9
Now, let D Ă P n be the divisor given by the union of n ´ general hyperplanes containing Π . For the stricttransform r D Ă X of D we have r D „ p n ´ qp H ´ E ´ ¨ ¨ ¨ ´ E q , and ´ K X ´ ǫ r D „ p n ` ´ ǫ p n ´ qq H ´ pp n ´ q ´ ǫ p n ´ qqp E ` . . . E q where H is the pull-back of the hyperplane section of P n via the blow-up morphism. Now, note that p´ K X ´ ǫ r D q¨p l ´ e i ´ e j q “ ǫ p n ´ q´ n ` , p´ K X ´ ǫ r D q¨p l ´ e i ´¨ ¨ ¨´ e q “ ǫ p n ´ q´ n ` , and p´ K X ´ ǫ r D q¨ e i “ n ´ ´ ǫ p n ´ q .Therefore, if n ě we have that ´ K X ´ ǫ r D is ample for any n ´ n ´ ă ǫ n ´ n ´ .Furthermore, since r D is the union of n ´ smooth irreducible divisors intersecting transversally along thesurface r Π the pair p X, ǫ r D q is klt for any ă ǫ ă . Then, for any n ´ n ´ ă ǫ ă the divisor ǫ r D induces a logFano structure on X , and [BCHM10, Corollary 1.3.2] yields that X is a Mori dream space.Now, consider the divisor D i „ H ´ E ´ ¨ ¨ ¨ ´ p E i ´ ¨ ¨ ¨ ´ E where the hat means that E i does no appearin the expression of D i . Note that D i is nef, and since D ni ą it is also big. Therefore, D i is semi-ample andbig. Now, consider a curve r C Ă X r C „ α , p l ´ e ´ e q ` ¨ ¨ ¨ ` α , p l ´ e ´ e q ` β r C ,..., ` γ e ` ¨ ¨ ¨ ` γ e where r C ,..., „ l ´ e ´ ¨ ¨ ¨ ´ e . Assume that D i ¨ r C “ γ ` ¨ ¨ ¨ ` γ “ . Hence we may write r C „ p β ` α , ` ¨ ¨ ¨ ` α , q l ´ p β ` α , ` ¨ ¨ ¨ ` α , q e ´ ¨ ¨ ¨ ´ p β ` α , q e Since β ` α , ` ¨ ¨ ¨ ` α , ă β ` α , ` ¨ ¨ ¨ ` α , ` ¨ ¨ ¨ ` β ` α , we conclude that r C Ă r Π .Then, a large enough multiple of D i induces a birational morphism f D i : X Ñ X i contracting r Π onto a P contained in X i , and whose exceptional locus is exactly r Π , that is Exc p f i q “ r Π . Indeed, D i | r Π yields the fibration r Π Ñ P induced by the linear system of conics in Π through p , . . . , p p i , . . . , p .Let f D i : X Ñ X i and f D j : X Ñ X j be the morphisms induced respectively by D i and D j . From now onwe will assume that n ě so that f D i is a small contraction. Then Exc p f D i q “ Exc p f D j q “ r Π . On the otherhand, D i and D j give rise to two different flops X ´ X X ` X i X jχ ´ f Di χ ` f Dj Therefore, X ` and X ´ correspond to two different chambers C ` and C ´ of the Mori chamber decomposition of Mov p X q . On the other hand, by Nakamaye’s theorem [Laz04, Theorem 10.3.5] the base locus of χ ` is Exc p f D j q and the base locus of χ ´ is Exc p f D i q . These base loci are respectively the stable base loci of D i and D j .Finally, since Exc p f D i q “ Exc p f D j q “ r Π we conclude that C ´ Y C ` is a unique chamber of the stable baselocus decomposition of Mov p X q .More generally, for i “ , . . . , we get five different chambers, all of them adjacent to Nef p X q , of the Morichamber decomposition of Mov p X q whose union gives a single chamber of its stable base locus decomposition.Furthermore, the stable base locus of a divisor in this chamber is exactly the surface r Π . Example 3.2. (Cox rings viewpoint) Let X be the toric variety with Cox ring R p X q : “ K r T , . . . , T s whosegrading matrix and irrelevant ideal are the following Q “ „ ´ ´
10 1 ´ ´ J irr p X q “ x T , T y X x T , T y X x T , T y The degrees of the generators are displayed in the following picture together with the semi-ample cone λ A whichis the gray region w w w w w λ A The matrix Q defines the following exact sequence where the right hand side Z is identified with the divisorclass group of X Ñ Z Ñ Z Q ÝÑ Z Ñ Denote by γ the positive quadrant of Q . Since the Cox ring is a polynomial ring the F -faces are all the faces of γ and the orbit cones are all their projections in Q . It follows that the maximal GIT chambers of X are all the -dimensional cones generated by pairs of consecutive rays. Recall that λ A is the GIT chamber correspondingto the semi-ample cone, generated by w , w . The corresponding bunch of orbit cones is Φ p λ A q : “ t orbit cones ω such that ω ˝ Ě λ ˝ A u In particular the only orbit cone of Φ which contains w is the whole of Q . Denote by λ i,j the cone determinedby ω i and ω j , and observe that cone p λ A Y λ , q “ cone p λ A Y λ , q “ Q . Thus, by Proposition 2.12, we concludethat λ sbl p w q “ λ , Y λ , . Example 3.3. (Fans viewpoint) Let v . . . , v P Z be a set of vectors which is Gale dual to the set w , . . . , w P Z in Example 3.2. We can assume v . . . , v to be the five columns of the following matrix »– ´ ´
10 1 0 1 20 0 1 1 0 fifl These vectors generate the one dimensional cones of a fan Σ whose cones are displayed in the following picture v v v v v The toric variety X “ X p Σ q is the same as the one previously defined in Example 3.2. The displayed fanstructure correspond to choosing the semi-ample chamber to be the GIT chamber λ A of the previous picture.The following are the fan structures of the toric varieties whose semi-ample cone is respectively cone p w , w q and cone p w , w q . v v v v v v v v v v Example 3.4. (Compactification) Let Y be the toric variety with Cox ring R : “ K r T , . . . , T s whose gradingmatrix and irrelevant ideal are the following Q “ »– fifl J irr p Y q “x T , T y X x T , T y X x T , T yX x T , T , T y X x T , T , T y The toric variety Y is a completion of the variety X in Example 3.2 obtained by adding the vector p , ´ , ´ q to the primitive generators of the one dimensional cones of X . The maximal cones of Y have the followingindexes: r , , s , r , , s , r , , s , r , , s , r , , s , r , , s , r , , s , r , , s . The variety Y is Q -factorial, non-Gorenstein, with K Y Cartier. The effective cone of Y has maximal GIT chambers, and the moving cone of N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 11 Y has maximal chambers. The columns of each of the following matrices generate a maximal GIT chamberof the moving cone, where the first one corresponds to the semi-ample cone A “ »– fifl M “ »– fifl M “ »– fifl The columns of each of the following matrices generate a maximal GIT chamber of the effective cone of X . C “ »– fifl C “ »– fifl The stable base locus of a divisor which lies in the interior of either C or C is V p T q . Thus the union of thesetwo chambers is a unique stable base locus chamber. On the other hand, if we denote by X i the projective toricvariety whose semi-ample cone is given by M i , then from the point of view of X i the GIT chambers coincidewith the stable base locus chambers. Example 3.5.
Consider the toric variety Z “ Z p Λ q with fan Λ Ă Q given in the following picture v v v v v v One can take for instance v , . . . , v to be the columns of the following matrix »– ´ ´ ´ ´ ´ fifl Then Z is a non-complete Mori dream space -fold with Picard number three and with Cox ring generated bysix free variables with degrees given by the columns w , . . . , w of the following matrix »– ´ ´ ´ ´ ´ fifl The effective cone of Z is the whole of Q , it has maximal GIT chambers and six of them are in the movablecone. Let us denote by Z j : “ Z p λ j q , j “ , . . . , the different models where Z “ Z , and λ j , j “ , . . . , arethe corresponding GIT chambers. The related fans are the following Λ Λ Λ Λ Λ Λ and the possible flips are as follows Z Z Z Z Z Z η η η η η η In Z we have that λ , λ , λ are in distinct stable base locus chambers but λ and λ are inside the same stablebase locus chamber. More precisely, if w j P λ ˝ j , j “ , . . . , then the stable base loci are B p w q “ H , B p w q “ l , , B p w q “ l , Y l , , B p w q “ l , Y l , , B p w q “ B p w q “ l , Y l , Y l , where l i,j is the curve in the intersection of the toric divisors D i , D j corresponding to the vectors v i , v j in thefan Λ .Geometrically we see that the flip η has l , as flipping curve, and l , does not exist in Z . Therefore, theflipping curve in the last flip is not visible from the point of view of Z .Note that we can produce an example of a complete -fold W with Picard number four such that MCD p W q ‰ SBLD p W q taking v “ ´ v ´ ¨ ¨ ¨ ´ v “ p , , ´ q and including the relevant additional cones.The following will be the leading example in the Section 5. Indeed, we will produce a complete Mori dreamspace X of Picard rank two such that MCD p X q ‰ SBLD p X q . Example 3.6. (Fundamental example) Let Z be the toric variety with Cox ring K r T , . . . , T s whose gradingmatrix and irrelevant ideal are the following Q “ „ J irr p Z q “ x T , T y X x T , . . . , T y Denote by w i P Z the degree of T i . The following picture displays the degrees of the generators of the Cox ringtogether with the three maximal GIT chambers of the moving cone, where the shaded one is the ample cone w , w w , . . . , w w , . . . , w w λ λλ A Let Z “ K be the spectrum of the Cox ring of Z and let X be the affine subvariety defined by X “ t F “ G “ u Ă Z where F, G are general polynomial of degree p , q in the T i . that is general linear combinations of the followingmonomials(3.7) $’’&’’% T T T T T T T T T T T T T T T T ; T T T T T T T T T T T T T T T T ; T T T T T T T T T T T T T T T T ; T T T T . It is immediate to check that for
F, G general enough X is irreducible and codim X p Sing p X qq ě . Since a localcomplete intersection is Cohen-Macaulay by Serre’s criterion on normality we get that X is a normal variety.Let p Z : p Z Ñ Z be the characteristic space morphism of Z and let p X : “ X X p Z . The image of p X via p Z isa subvariety X of Z . Since X is irreducible and normal, and X is a GIT quotient of X by a reductive group[Bri10, Theorem 1.24 (vi)] yields that X is irreducible and normal as well. We claim that p X “ Zariski closure of p ´ Z p X X Z q in p Z N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 13 where Z is the smooth locus of Z . Indeed Z contains the open subset Z of Z obtained by removing the unionof all the toric subvarieties of the form p Z p V p T i , T j qq , for any pair of indexes i, j . Since the Zariski closure of p ´ Z p X X Z q in p Z equals p X the claim follows.Observe that codim X p X z p X q ě . Thus if one can show that the pull-back ı ˚ : Cl p Z q Ñ Cl p X q , induced bythe inclusion, is an isomorphism then by [ADHL15, Corollary 4.1.1.5], it follows that the Cox ring of X is R p X q “ K r T , . . . , T s I p X q Note that each of w .. , w .. and w .. is an orbit cone of X . In particular, the three maximal chambers λ , λ , λ A of the moving cone are GIT chambers. On the other hand, since w , , is not an orbit cone, it follows thateach orbit cone contains t ω : λ A Ď ω and λ Ř ω u “ t ω : λ A Ď ω and λ Ř ω u Then λ and λ are contained in the same SBL chamber. It remains to show that ı ˚ : Cl p Z q Ñ Cl p X q is an isomorphism. Note that the K ˚ ˆ K ˚ action on X zp V p T q Y V p T qq is trivial, so if we remove the imagesof V p T q Y V p T q from X , the resulting variety is isomorphic to an affine space. Therefore, Cl p X q is generatedby the classes of the images of the two irreducible divisors V p T i q X X , with i P t , u , and ρ p X q ď .Note that crossing the wall corresponding to w , w we get a morphism f : Z Ñ P . Furthermore, X Ă Z is not contained in any fiber of f , and hence f restricts to a surjective morphism f | X : X Ñ P . This forces ρ p X q ě . Finally, we conclude that the images of V p T q , V p T q form a basis of Cl p X q and ρ p X q “ .We finish the present section with two examples of toric complete varieties in which the Mori chamber andstable base locus decomposition do not coincide even inside the movable cone. Example 3.8.
Let X be a toric variety with the following grading matrix Q : “ »– fifl This is a toric -fold whose Mori chamber and stable base locus decomposition do not coincide even insidethe movable cone. More precisely, its effective cone has GIT chambers, of these are inside the movablecone. The picture below shows a section of the Mori chamber decomposition of Eff p X q . The inner chambers,two triangles and three quadrilaterals, are the GIT chambers of the movable cone. There are five possiblemodels, the picture shows in black a chosen semi-ample cone, and in the corresponding model the two gray GITchambers C and C share the stable base locus.If one chooses any of the other four models the situation is similar, the two GIT chambers opposite to thechosen semi-ample cone inside the movable cone will have the same stable base locus. Example 3.9.
Let X be a toric variety with the following grading matrix Q : “ »– fifl This is a toric -fold whose Mori chamber and stable base locus decomposition do not coincide inside themovable cone. More precisely, its effective cone has GIT chambers, all of them inside the movable cone since Eff p X q “ Mov p X q . The picture below shows a section of the Mori chamber decomposition of Eff p X q . There arethree possible models, the picture shows in black a chosen semi-ample cone, and in the corresponding model the two gray GIT chambers C and C have the stable base locus. If one chooses any of the other two models thesituation is the same. Remark 3.10.
All the computations of this section have been implemented in Magma [BCP97] and Maple[HK15] programs. For convenience of the reader we include, as ancillary files in the arXiv version of the paper,the following files:-
Readme.txt , a text on how to use the remaining files;-
SBLib.m , the Magma library containing all the functions needed to verify our examples;-
Examples.txt , the examples.For an optimized version, implemented in Maple and Singular, of some of the algorithms presented in thislibrary see [Kei12]. The library
SBLib.m contains nine commands which we briefly describe here.-
Ffaces : computes the F -faces of an ideal I of a polynomial ring according to [ADHL15, Remark 3.1.1.11].- Eff : computes the effective cone of a family of vectors in a rational vector space according to [ADHL15,Definition 2.2.2.5]. It takes as input the grading matrix whose columns are the relevant vectors.-
Mov : computes the moving cone of a family of vectors in a rational vector space according to [ADHL15,Definition 2.2.2.5]. It takes as input the grading matrix whose columns are the named vectors.-
OrbitCones : computes the orbit cones as projections of the F -faces according to [ADHL15, Proposition3.1.1.10]. It takes as input a pair consisting of the set of F -faces together with the grading matrix.- GitChamber : computes the GIT chamber defined by a class w according to [ADHL15, Definition 3.1.2.6].It takes as input a pair consisting of the set of orbit cones together with a class w .- GitFan : computes the GIT quasi-fan, that is the collection of all the git cones, of the set of orbit conesaccording to [Kei12, Algorithm 8]. It takes as input a pair consisting of the set of orbit cones together witha class w .- BunchCones : computes a bunch of orbit cones defined by a GIT chamber λ according to [ADHL15, Example3.1.3.6]. It takes as input a pair consisting of the set of orbit cones together with a class w in the relativeinterior of λ .- SameSbl : decides whether two classes w , w have the same stable base locus according to Proposition 2.8.It takes as input a triple consisting of a bunch of orbit cones together with the two classes w and w .- FindTriples : determines all the triples p λ A , λ , λ q of GIT chambers such that λ and λ are contained inthe same stable base locus chamber of the variety whose semi-ample chamber is λ A . It works according toProposition 2.8, and takes as input a triple consisting of the grading matrix, the set of orbit cones and theGIT fan. 4. Smooth toric 3-folds of Picard rank three
In this section we prove Theorem 1.2. Let N be a finitely generated free abelian group, M : “ Hom p N, Z q , N Q : “ N b Z Q , and let X : “ X p Σ q be a smooth projective toric variety defined by a fan Σ Ď N Q . Since X isprojective Σ is the normal fan of the Riemann-Roch polytope ∆ Ď M Q of any ample divisor of X , in particulareach facet of ∆ corresponds to a one dimensional cone of Σ . Since X is smooth each maximal cone of X issimplicial so that each vertex of ∆ has valence n : “ dim p X q .When X is a threefold of Picard rank three the polytope ∆ has six facets and all its vertexes have valencythree. If we denote by p v, e, f q the vector consisting of the number of vertexes, edges and facets of ∆ , thenthe conditions f “ , e “ v and v ´ e ` f “ give p v, e, f q “ p , , q . According to the classification ofhexahedra there are only two possible topological types for ∆ , displayed in the following pictures. N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 15
Number the facets of each polytope from to and let C i be the primitive generator of the inward normalvector to the i -th facet. Any vertex is labeled by a triple p i, j, k q in such a way that(4.0) det p C i , C j , C k q “ . For the type I polytope the vertexes labels are: p , , q , p , , q , p , , q , p , , q , p , , q , p , , q , p , , q , p , , q , while for the type II polytope the vertexes labels are: p , , q , p , , q , p , , q , p , , q , p , , q , p , , q , p , , q , p , , q . Without loss of generality we can assume t C , C , C u to be the canonical basis of N Q » Q . Thus the matrix whose columns are the primitive generators of the fan Σ and its orthogonal havethe form P : “ »– a a a b b b c c c fifl Q : “ »– ´ a ´ b ´ c ´ a ´ b ´ c ´ a ´ b ´ c fifl Applying conditions (4.0) to all type I triples gives the following equations a “ b “ c “ ´ , a c “ b c “ a b “ , a b c ` a b c “ . These equations cut out the union of six three-dimensional affine spaceswhich are in a unique orbit of the S action which permutes the coordinates. One of these spaces is given by a “ b “ c “ ´ , b “ c “ c “ . The corresponding Q matrix is Q : “ »– ´ a ´ a ´ b fifl Any integer vector p a , a , b q gives an example of a smooth toric threefold whose defining fan is of type I.Applying conditions (4.0) to all the type II triples gives the following equations a “ b “ c “ ´ , b c “ a c “ , a b ` a ` “ , b ` a b ` “ . These equations cut out the union of three irreduciblesubvarieties of dimension two. Two of these are the affine spaces a “ b “ b “ c “ ´ , a “ c “ , a “ and a “ b “ a “ c “ ´ , b “ c “ , b “ . The Q matrix for points on the first affine space is Q : “ »– ´ b ´ c ´ fifl The second affine space gives a similar Q matrix. The third variety has equations a “ b “ c “ ´ , c “ c “ , a b ` a ` “ , a b ` b ` “ . Considering only integer points this third variety is the unionof a point and two one parameter families corresponding to the following grading matrices Q : “ »– ´ fifl , Q : “ »– ´ ´ a fifl , Q : “ »– ´ a ´ a ` fifl . Note now that the case Q is a sub-case of Q and the case Q is a sub-case of Q . Therefore it is enoughto analyze Q , Q and Q . We will call
A, B, . . . , F the columns of Q. Note that in all cases D “ p , , q , E “p , , q and F “ p , , q . In the case of Q we have C “ F and the point B moves in the half line from F to E . There are threepossibilities b ă , b “ , b ą . If b “ there is nine possibilities for the point A according to a and a being negative, zero or positive. If b ă we now have four more possibilities depending if the point A is aboveor below the line DF, therefore possibilities. For b ą we have possibilities as well. Summing up wehave possibilities. Choosing a particular value of b , a , a for each of the cases and computing the Mori chamber and the stable base locus decomposition one gets exactly two cases where the Mori chamber and stablebase locus decomposition do not coincide, corresponding to the following matrices G : “ »– α β fifl , α, β ą , G : “ »– α β γ fifl , α, β, γ ă Now, consider the case Q . In this case A is fixed, B moves in the line connecting p , , q to D and C movesin the line DF. If c “ then C “ D and B has three possibilities according to sign of b . If c ă there arefive possibilities for b depending on the sign and on which of the inequalities b ě c and b ă c holds. For c ą there are five possibilities as well but now what matters is the sign of b and if it is greater than c ` or not. Therefore there are possibilities to check. Computing explicitly the decompositions we get newpossible types of varieties whose Mori chamber and stable base locus decompositions do not coincide: G : “ »– γ ´ fifl , γ ą , G : “ »– β γ ´ fifl , β ą ą γG : “ »– β γ ´ fifl , β ă ă γ, G : “ »– β γ ´ fifl , β ă γ ´ ă ´ Finally, doing the same with Q we have to consider four cases: a ă ( A inside the triangle DEF ), a “ ( A in the midpoint of CF ), a “ ( A in the line BF ), and a ą ( A below the line BF ). Doing this we get asingle new type: G : “ »– ´ fifl This proves Theorem 1.2. 5.
Mori dream spaces of Picard rank two
Let X be a Mori dream space with divisor class group Cl p X q of rank two. Since X is a projective variety itseffective cone is pointed. Moreover Cl p X q has rank two so that we can fix a total order on the classes in theeffective cone w ď w if w is on the left of w . Given two convex cones λ, λ contained in the effective cone wewill write λ ď λ if w ď w for any w P λ and w P λ Denote by t f , . . . , f r u a minimal set of homogeneous generators for the Cox ring R p X q of X , and let w i “ deg p f i q for any i . Proposition 5.1.
Let X be a projective Q -factorial toric variety with Picard rank two. Then the Mori chamberand the stable base locus decompositions of Eff p X q coincide.Proof. Let λ A be the semi-ample cone of X and let λ , λ be two distinct maximal GIT chambers of X .According to (2.10) it suffices to show that there exists an orbit cone of the bunch which contains one of λ A Y λ , λ A Y λ but not the other. Since X is complete the effective cone is pointed, so that, since the Picardrank is two, one can order the GIT chambers of X .Assuming λ ď λ , we have three possibilities: either λ ď λ A ď λ , or λ A ď λ ď λ , or λ ď λ ď λ A . Since X is toric each pair of degrees of generators of the Cox ring span an orbit cone. Thus in the first two cases wecan find an orbit cone which contains λ A Y λ but not λ A Y λ , while in the last case we can find an orbit conewhich contains λ A Y λ but not λ A Y λ . (cid:3) The following is our first simple criterion implying the equality of the two chamber decompositions for a Q -factorial Mori dream space of Picard rank two. Proposition 5.2.
Let X be a projective Q -factorial Mori dream space with Picard rank two. If all the generatorsof R p X q appear in the walls of the stable base locus decomposition of Eff p X q then the Mori chamber and thestable base locus decompositions of Eff p X q coincide. N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 17
Proof.
By (2.7) the Mori chamber decomposition is a subdivision of Eff p X q whose walls are given by some ofthe generators of R p X q , and it is a refinement of the stable base locus decomposition. Since, by hypothesis allthe generators appear as walls of the stable base locus decomposition such a refinement must be trivial. (cid:3) Now, we develop some technical results in order to describe the semi-stable loci corresponding to the GITchambers of the Mori chamber decomposition of a Mori dream space of Picard rank two.
Lemma 5.3.
Let X be a Mori dream space with Picard rank two, λ Ď Cl Q p X q a maximal GIT chamber of X ,and X ss p λ q the corresponding subset of semi-stable points of X . Then the following holds X z X ss p λ q “ V p f i : w i ď λ q Y V p f i : λ ď w i q Proof.
By [ADHL15, Theorem 3.1.2.8] we have X ss p λ q “ X ss p w q for any w P λ ˝ , where the second semi-stablelocus is the complement of the zero set of all the homogeneous sections of the Cox ring whose degree is a positivemultiple of w . If we choose such a class w P λ ˝ so that w ă w i for any w i P λ ˝ then each monomial in f , . . . , f r of degree nw must contain at least one f i with w i ď λ . Thus the inclusion V p f i : w i ď λ q Ď X z X ss p λ q follows. The analogous inclusion for V p f i : λ ď w i q can be proved in a similar way.To prove the opposite inclusion observe that if x P X is a point which does not belong to the union V p f i : w i ď λ q Y V p f i : λ ď w i q , then there exist two sections f i , with w i ď λ , and f j , with λ ď w j , each of whichdoes not vanish on x .Take non negative a, b P Z such that aw i ` bw j P λ ˝ . Since f ai f bj is a homogeneous element of the Cox ringof degree aw i ` bw j P λ ˝ which does not vanish on x , the point x is in X ss p λ q . (cid:3) Lemma 5.4.
Let X be a Mori dream space with Picard rank two and let λ, λ Ď Cl Q p X q be two maximal distinctGIT chambers of X with λ ď λ . Then the following inclusion is strict V p f i : w i ď λ q Ř V p f i : w i ď λ q Proof.
Assume that the equality V p f i : w i ď λ q “ V p f i : w i ď λ q holds. By hypothesis the inclusion V p f i : λ ď w i q Ď V p f i : λ ď w i q holds. Thus by Lemma 5.3 there would be an inclusion X ss p λ q Ď X ss p λ q .By [ADHL15, Theorem 3.1.2.8] the latter inclusion would imply λ Ď λ , a contradiction. (cid:3) Recall that a Mori dream space X is a good quotient of its characteristic space p X “ X ss p λ A q , and denoteby p λ A : p X Ñ X the good quotient map. The following simple characterization of stable base loci will befundamental for the rest of the paper. Lemma 5.5. If λ ď λ A then the stable base locus of a class w P λ is (5.6) B p w q “ p λ A p p X z X ss p λ qq “ p λ A p p X X V p f i : w i ď λ qq If λ ě λ A then the stable base locus of a class w P λ is (5.7) B p w q “ p λ A p p X z X ss p λ qq “ p λ A p p X X V p f i : w i ě λ qq Proof.
In order to prove (5.6) just note that the first equality holds by definition while the second equalityis due to Lemma 5.3 and the fact that λ ď λ A . Clearly (5.7) can be proved using a completely analogousargument. (cid:3) The following is the main technical tool of the paper.
Theorem 5.8.
Let X “ X p λ A q be a Q -factorial Mori dream space with Picard rank two corresponding to themaximal chamber λ A of the Mori chamber decomposition of Eff p X q . If for any λ ď λ ď λ A we have (5.9) V p f i : w i ď λ qz V p f i : w i ě λ A q Ř V p f i : w i ď λ qz V p f i : w i ě λ A q and for any λ A ď λ ď λ we have (5.10) V p f i : w i ě λ qz V p f i : w i ď λ A q Ř V p f i : w i ě λ qz V p f i : w i ď λ A q then the Mori chamber and the stable base locus decompositions of Eff p X q coincide.Furthermore, if for any λ ď λ A ď λ we have that V p f i : w i ď λ qzp V p f i : w i ď λ A q Y V p f i : w i ě λ A qq is different from V p f i : w i ě λ qzp V p f i : w i ď λ A q Y V p f i : w i ě λ A qq then the stable base locus chambers of Eff p X q are convex.Proof. Let λ be a maximal non-ample GIT chamber of X . Assume that λ ď λ A , where λ A is the ample coneof X . By (5.6) in Lemma 5.5 the stable base locus of a class w P λ is B p w q “ p λ A p p X z X ss p λ qq “ p λ A p p X X V p f i : w i ď λ qq Now let λ ď λ A be any maximal GIT chamber distinct from λ and λ A . Without loss of generality we canassume that λ ď λ then by Lemma 5.4 we deduce the following V p f i : w i ď λ A q Ř V p f i : w i ď λ q Ř V p f i : w i ď λ q V p f i : w i ě λ A q Ś V p f i : w i ě λ q Ś V p f i : w i ě λ q where all the inclusions are strict. Taking the intersection with p X is equivalent to remove from V p f i : w i ď λ q Ĺ V p f i : w i ď λ q the common subset V p f i : w i ď λ A q and their intersection with V p f i : w i ě λ A q . Sohypothesis (5.9) yields that p X X V p f i : w i ď λ q ‰ p X X V p f i : w i ď λ q Since X is Q -factorial, the good quotient p λ A : p X Ñ X is geometric [ADHL15, Corollary 1.6.2.7]. It followsthat the images of the above sets via p λ A remain distinct in X and thus that B p w q ‰ B p w q for any w P λ ˝ and w P λ .Now, assume that λ A ď λ . Then (5.7) in Lemma 5.5 yields that the stable base locus of a class w P λ is B p w q “ p λ A p p X z X ss p λ qq “ p λ A p p X X V p f i : w i ě λ qq In this case if λ is a maximal chamber distinct from λ such that λ A ď λ ď λ Lemma 5.3 yields the followingstrict inclusions V p f i : w i ě λ A q Ř V p f i : w i ě λ q Ř V p f i : w i ě λ q V p f i : w i ď λ A q Ś V p f i : w i ď λ q Ś V p f i : w i ď λ q To conclude it is enough to argue as in the previous case using (5.10) instead of (5.9).Summing up we showed that any pair of distinct GIT chambers lying on the same side of λ A gives twodifferent stable base locus chambers. Therefore, the Mori chamber decomposition of Eff p X q coincide with itsstable base locus decomposition.Finally, an analogous argument shows that if λ ď λ A ď λ and our last hypothesis holds then for any w P λ and w P λ we have B p w q ‰ B p w q , and hence the stable base locus chambers of Eff p X q are convex. (cid:3) Remark 5.11.
Let us consider the Mori dream space X in Example 3.6. Note that (3.7) yields V p f i : w i ě λ A q “ t T “ T “ r F “ r G “ u where r F , r G are general linear combinations of the following monomials " T T T T T T T T T T T T T T T T ; T T T T T T T T T T T T T T T T ; V p f i : w i ď λ q “ t T “ T “ T “ T “ T “ u V p f i : w i ď λ q “ t T “ T “ T “ T “ T “ T “ u Y t T “ T “ T “ T “ T “ u and V p f i : w i ě λ A q Ą t T “ T “ T “ T “ T “ T “ u contains a component of the set V p f i : w i ď λ q .In what follows we work out some interesting consequences of Theorem 5.8. Corollary 5.13.
Let X “ X p λ A q be a Q -factorial Mori dream space with Picard rank two corresponding tothe maximal chamber λ A of the Mori chamber decomposition of Eff p X q . If for any maximal chamber λ wehave that V p f i : w i ď λ q and V p f i : w i ě λ q are irreducible then the Mori chamber and the stable base locusdecompositions of Eff p X q coincide. N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 19
Proof.
Without loss of generality we may assume that λ ď λ ď λ A . Since V p f i : w i ď λ q is irreducible either V p f i : w i ě λ A q Ě V p f i : w i ď λ q or V p f i : w i ě λ A q X V p f i : w i ď λ q is a closed subset of V p f i : w i ď λ q .Assume that V p f i : w i ě λ A q Ě V p f i : w i ď λ q . Then since V p f i : w i ě λ q Ď V p f i : w i ě λ A q weget that X ss p λ A q Ď X ss p λ q , and [ADHL15, Theorem 3.1.2.8] yields that λ Ď λ A , a contradiction. Therefore, V p f i : w i ě λ A q intersects V p f i : w i ď λ q in a closed subset, and to conclude it is enough to apply Theorem5.8. (cid:3) Now, we are ready to prove the main result of the paper.
Theorem 5.14.
Let X be a Q -factorial Mori dream space with Picard rank two, t f , . . . , f r u a minimal set ofhomogeneous generators for the Cox ring R p X q , w i : “ deg p f i q , and λ A be the ample chamber of X . Denote by c the codimension of X into its canonical toric embedding [ADHL15, Section 3.2.5] . Define h ` : “ t f i : w i ě λ A u and h ´ : “ t f i : w i ď λ A u If h ´ ą c and h ` ą c , then the Mori chamber and the stable base locus decomposition of Eff p X q coincide.Proof. Consider p λ A : p X Ñ X , let X be the total space of X , and Z – A r be the affine space with coordinatesgiven by the f i . Let λ , λ be two chambers of the Mori chamber decomposition of Eff p X q as in the followingpicture λ λλ A Recall that by (5.6) in Lemma 5.5 the stable base loci of classes w P λ , w P λ are given respectively by B p w q “ p λ A p p X z X ss p λ qq “ p λ A p p X X V p f i : w i ď λ qq B p w q “ p λ A p p X z X ss p λ qq “ p λ A p p X X V p f i : w i ď λ qq and the non semi-stable locus of λ A is V p f i : w i ď λ A q Y V p f i : w i ě λ A q Now, X Ă A r has dimension dim p X q ` , and hence any irreducible component of the intersection X X V p f i : w i ď λ q has dimension greater than or equal to dim p X q ` ´ h , where h “ t f i : w i ď λ u . Assume thatan irreducible component of X X V p f i : w i ď λ q is contained in X X V p f i : w i ě λ A q . Then such componentmust be contained in V p f i , f j : w i ď λ , w j ě λ A q which has dimension r ´ h ´ h ` . This forces h ` ď c , a contradiction with our hypothesis. Now, to concludethat λ, λ are two different stable base locus chambers it is enough to recall that Lemma 5.4 yields V p f i : w i ď λ q Š V p f i : w i ď λ q .When λ A ď λ ď λ we argue in a completely analogous way, and then to conclude it is enough to applyTheorem 5.8. (cid:3) The following is the first immediate consequence of Theorem 5.14.
Corollary 5.15.
Let Z be a projective normal Q -factorial toric variety with rk p Cl p Z qq “ , and X Ď Z aprojective normal Q -factorial Mori dream hypersurface such that ı ˚ : Cl p Z q Ñ Cl p X q is an isomorphism. Thenthe Mori chamber and the stable base locus decompositions of both Eff p Z q and Eff p X q coincide.Proof. For a toric variety the claim follows from Proposition 5.1 and it is also an immediate consequence ofTheorem 5.14 with c “ . In general, following the notation in the proof of Theorem 5.14, there are alwaysat least two generators in the sets t f i : w i ě λ A u , t f i : w i ď λ A u otherwise λ A would be a chamber of Eff p X qz Mov p X q . Since c “ codim Z p X q “ we conclude by Theorem 5.14. (cid:3) Remark 5.16.
Theorem 5.14 is sharp. Indeed, the Mori dream space in Example 3.6 has three Mori chamberbut just two stable base locus chambers. In this example h ` “ c “ . Remark 5.17.
An intrinsic quadric is a normal Q -factorial projective Mori dream space with Cox ring definedby a single quadratic relation. Smooth intrinsic quadrics with small Picard rank have been studied recently in[FH18]. By Corollary 5.15 the Mori chamber and the stable base locus decomposition of the effective cone ofan intrinsic quadric of Picard rank two coincide.5.17. Picard rank two varieties with a torus action of complexity one.
Recall that a variety with atorus action of complexity one is a normal complete algebraic variety X with an effective action of a torus T such that the biggest T -orbits are of codimension one in X . Proposition 5.18.
Let X be smooth rational projective variety of Picard rank two that admits a torus actionof complexity one. Then the Mori chamber and the stable base locus decomposition of Eff p X q coincide.Proof. By [FHN16, Theorem 1.1] any smooth rational projective variety of Picard rank two with a torus actionof complexity one, with just one exception, is a Mori dream hypersurface in its canonical toric embedding.Therefore, with the exception of the variety No. 13 in the statement of [FHN16, Theorem 1.1] the claim followsdirectly from Corollary 5.15.On the hand, the Cox ring of the exceptional variety X has eight generators T , . . . , T , with deg p T q “ deg p T q “ deg p T q “ deg p T q , and deg p T q “ deg p T q “ deg p T q “ deg p T q . Therefore, both the Mori chamberand the stable base locus decomposition of Eff p X q consist of a single chamber which is indeed the nef cone of X . (cid:3) In what follows we apply the techniques developed in this section to compute the stable base locus decom-position which by Proposition 5.18 coincide with the Mori chamber decomposition, of the effective cones of thevarieties in [FHN16, Theorem 1.1].
Example 5.19. (No. 6 in [FHN16, Theorem 1.1]) In this case X is a variety of dimension m ` with Cox ringgiven by R p X q – k r T , . . . , T , S , . . . , S m sp T T ` T T ` T T q with m ě , and grading matrix ˆ c ` a b c . . .
11 1 1 1 1 0 0 . . . ˙ with a, b, c ě , a ă b and a ` b “ c ` . Here we develop the case ă a ă c , when a “ or a “ c a similarargument will work. Therefore, MCD p X q is a possibly trivial coarsening of the following decomposition w w w w w S , . . . , S m λ λ λ λ λ where λ “ λ A is the ample cone of X . Note that (5.6) in Lemma 5.5 yields B p w q “ p λ A p p X X V p f i : w i ď λ qq “ p λ A p p X X t T “ T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ď λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ď λ qq “ p λ A p p X X t T “ T “ T T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ď λ qq “ p λ A p p X X t T “ T T ` T T “ uq if w P λ . Therefore,
MCD p X q “ SBLD p X q “ t λ A , λ , λ , λ , λ u . Example 5.20. (No. 8 in [FHN16, Theorem 1.1]) In this case X is a variety of dimension m ` with Cox ringgiven by R p X q – k r T , . . . , T , S , . . . , S m sp T T ` T T ` T T q N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 21 with m ě , and grading matrix ˆ . . .
11 1 1 1 1 1 0 a . . . a m ˙ with ď a ď ¨ ¨ ¨ ď a m and a m ą . We develop the case ă a ă ¨ ¨ ¨ ă a m , the same argument will work inthe remaining cases as well. Therefore, MCD p X q is a possibly trivial coarsening of the following decomposition w w w m ` w m ` w , . . . , w λ λ m ´ λ m where λ m “ λ A is the ample cone of X . Note that (5.7) in Lemma 5.5 yields B p w q “ p λ A p p X X V p f i : w i ě λ j qq “ p λ A p p X X t S j “ ¨ ¨ ¨ “ S “ T T ` T T ` T T “ uq if w P λ j , for j “ , . . . , m ´ . Therefore, MCD p X q “ SBLD p X q “ t λ A , λ m ´ , . . . , λ u .For all the other varieties listed in [FHN16, Theorem 1.1], with the exception of the varieties No. 3 andNo. 12 for generic parameters, arguing similarly we get that the Mori chamber decomposition of the varietycoincide with the one of the ambient toric variety which is given by the corresponding grading matrix in [FHN16,Theorem 1.1]. In the following we study the two exceptional cases. Example 5.21. (No. 3 in [FHN16, Theorem 1.1]) In this case X is a -fold with Cox ring given by R p X q – k r T , . . . , T sp T T T ` T T ` T q with m ě , and grading matrix ˆ ´ a a ˙ with a ě . Therefore, in the case a ě the MCD p X q is a possibly trivial coarsening of the followingdecomposition w w w w w , w λ λ λ λ where λ “ λ A is the ample cone of X . Note that (5.7) in Lemma 5.5 yields B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ uq if w P λ . Therefore,
MCD p X q “ SBLD p X q “ t λ A , λ Y λ , λ u .In the case a “ , MCD p X q is a possibly trivial coarsening of the following decomposition w , w w w w , w λ λ λ where λ “ λ A is the ample cone of X . Note that (5.7) in Lemma 5.5 yields B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ . Therefore,
MCD p X q “ SBLD p X q “ t λ A , λ Y λ u .In the case a “ , MCD p X q is a possibly trivial coarsening of the following decomposition w w , w , w w , w λ λ where λ “ λ A is the ample cone of X . Now (5.7) in Lemma 5.5 yields B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ uq if w P λ . Therefore,
MCD p X q “ SBLD p X q “ t λ A , λ u . Example 5.22. (No. 12 in [FHN16, Theorem 1.1]) In this case X is a variety of dimension m ` with Coxring given by R p X q – k r T , . . . , T , S , . . . , S m sp T T ` T T ` T q with m ě , and grading matrix ˆ . . .
00 2 c a b c . . . ˙ with ď a ď c ď b and a ` b “ c . Therefore, in the case ă a ă c ă b , MCD p X q is a possibly trivialcoarsening of the following decomposition w w w w w w m ` , . . . , w m ` λ λ λ λ λ where λ “ λ A is the ample cone of X . Note that (5.7) in Lemma 5.5 yields B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ ; B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T “ T “ uq if w P λ . B p w q “ p λ A p p X X V p f i : w i ě λ qq “ p λ A p p X X t T “ T T ` T “ uq if w P λ . Therefore,
MCD p X q “ SBLD p X q “ t λ A , λ , λ Y λ , λ u .If there is an equality in any of the inequalities ď a ď c ď b , then some w j coincide and the correspondingchambers collapse as in Example 5.21. For instance, if a “ then w “ w and the chamber λ does not exist.6. Grassmannians blow-ups
Let G p r, n q be the Grassmannian parametrizing r -planes in P n , and G p r, n q k the blow-up of P n at k generalpoints. These blow-ups have been studied in [MR18]. In particular the stable base locus decomposition of Eff p G p r, n q q has been computed in [MR18, Theorem 1.3].In this section we will compute the Cox ring of G p r, n q by exploiting its spherical nature, and as a consequenceof Proposition 5.2 we will answer positively to [MR18, Question 6.9] which ask whether the decomposition givenin [MR18, Theorem 1.3] is the Mori chamber decomposition of Eff p G p r, n q q . N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 23
Definition 6.1. A spherical variety is a normal variety X together with an action of a connected reductiveaffine algebraic group G , a Borel subgroup B Ď G , and a base point x P X such that the B -orbit of x in X is a dense open subset of X .Let p X, G , B , x q be a spherical variety. We distinguish two types of B -invariant prime divisors: a boundarydivisor of X is a G -invariant prime divisor on X , a color of X is a B -invariant prime divisor that is not G -invariant.For instance, any toric variety is a spherical variety with B “ G equal to the torus. For a toric variety thereare no colors, and the boundary divisors are the usual toric invariant divisors.Set Λ : “ t I Ă t , . . . , n u , | I | “ r ` u and N : “ | Λ | ` . Define the Hamming distance on Λ as d p I, J q “ | I | ´ | I X J | “ | J | ´ | I X J | for each I, J P Λ . Note that, with respect to this distance, the diameter of Λ is r ` . We consider theGrassmannian G p r, n q in the usual Plücker embedding G p r, n q Ă P N .For each pair I “ t i ă ¨ ¨ ¨ ă i r ´ u , J “ t j ă ¨ ¨ ¨ ă j r ` u Ă t , . . . , n u with | I | “ r, | J | “ r ` , define aquadratic polynomial(6.2) F IJ “ r ` ÿ t “ p´ q t p i ...i r ´ j t p j ... p j t ...j r ` Then the ideal of K r p I , I P Λ s generated by the F IJ is the ideal defining G p r, n q Ă P N [Sha13, Section I.4].We denote by G p r, n q the blow-up of G p r, n q at p “ x e , . . . , e r y , where t e , . . . , e n u is the canonical basis of K n ` , by H the pull-back to G p r, n q of the hyperplane section of P N , and by E the exceptional divisor of theblow-up. Proposition 6.3.
In the polynomial ring K r S, T I , I P Λ s consider the ideal J generated by the Plücker relations (6.2) in the coordinates T I . Then R p G p r, n q q – K r S, T I , I P Λ s J and the degree of the variable T I in Cl p G p r, n q q “ Z r H s ` Z r E s is p , ´ d p I, t , . . . , r uqq , where deg p S q “ p , q .Proof. By [MR18, Proposition 4.1] under the action of the reductive group G “ "ˆ A B ˙ , A P GL r ` , B P GL n ´ r | det p A q det p B q “ * Ă SL n ` the blow-up G p r, n q is a spherical variety. We consider the Borel subgroup B Ă G of matrices with uppertriangular blocks. Consider the divisors D , . . . , D r ` in G p r, n q defined as D j : “ tr Σ s P G p r, n q : Σ X Γ j ‰ ∅ u ,where $’’’’’’&’’’’’’% Γ “ x e r ` , . . . , e n y ;Γ “ x e , e r ` , . . . , e n ´ y ; ... Γ r ` “ x e , . . . , e r , e r ` , . . . , e n ´ r ´ y ;Γ “ x e , . . . , e r y . Pulling-back these divisors via the blow-up map we obtain divisors in G p r, n q . For sake of simplicity we willuse the same notation for divisors in G p r, n q and their pull-backs in G p r, n q .Now, note that G preserves the dimension of the intersection of a given subspace of P n with Γ and with Γ .Therefore G ¨ D “ D and G ¨ E “ E that is, D and E are boundary divisors. Note also that each D j is a B -invariant but not a G -invariant prime divisor, and therefore D , . . . , D r ` are colors.In order to determine the G -orbit of D , . . . , D r ` we have to describe these divisors explicitly as zeros ofpolynomials in the Plücker coordinates.In G p r, n q the divisor D is given by D “ V p p ,...,r q . Indeed, if q P D then q “ r Σ s with Σ X Γ ‰ ∅ andtherefore it can be represented with a matrix whose first row is of the form p , . . . , , a r ` , . . . , a n q .This implies that p ...r p q q “ . Conversely, if p ...r p q q “ then the most left p r ` q ˆ p r ` q sub-matrixof any matrix representing q has zero determinant, therefore there is another representation of q such that thefirst row has the following form p , . . . , , a r ` , . . . , a n q , and we conclude that q P D . Similarly, setting I j “ t e , . . . , e j ´ , e r ` , . . . , e n ´ j u for j “ , . . . , r ` , we have D j “ V p p I j q . Note that d p I j , I q “ j for any j . More generally, one can consider the prime divisor D I : “ V p p I q for any I P Λ . We claimthat the linear span of the orbit G ¨ D j is given by(6.4) lin p G ¨ D j q “ xt D J ; d p J, I q “ j uy , j “ , . . . , r ` Note that given
I, I P Λ with distance one and such that the non shared indexes, say i P I z I , i P I z I arein t , . . . , r u we can find a g P G such that g p e i q “ e i and g p e j q “ e j for j ‰ i, i . The same holds if the nonshared indexes are in t r ` , . . . , n u , and we get that t D J ; d p J, I q “ j u Ă G ¨ D j , j “ , . . . , r ` Since the D J such that d p J, I q “ j, j “ , . . . , r ` , give a generating set of H p G p r, n q , O G p r,n q p qq we get(6.4). Now, let S and T I be the canonical sections associated respectively to E and D I . By [ADHL15, Theorem4.5.4.6] S, T I , I P Λ are generators of R p G p r, n q q . Furthermore, by [Ris17, Lemma 7.2.1] for any I P Λ we havethat mult x e ,...,e r y D I “ ` dim px e i , i R I y X x e , . . . , e r yq “ |pt , . . . , n uz I q X t , . . . , r u| “ d p I, I q Therefore, if deg p S q “ p , q the degree of the other generators of R p G p r, n q q in Pic p G p r, n q q “ Z r H s ‘ Z r E s is given by deg p T I q “ p , ´ d p I, I qq .The matrix representing this grading has size ˆ p N ` q and is of the following form A “ ˆ . . . . . .
11 0 ´ . . . ´ ´ . . . ´p r ` q ˙ Our next aim is to find relations among the generators of R p G p r, n q q . Note that for each pair I “ t i ă ¨ ¨ ¨ ă i r ´ u , J “ t j ă ¨ ¨ ¨ ă j r ` u Ă t , . . . , n u with | I | “ r, | J | “ r ` , the polynomial G IJ “ r ` ÿ t “ p´ q t T i ...i r ´ j t T j ... p j t ...j r ` is homogeneous of degree p , ´| I z I | ´ | J z I |q . Let J Ă K r T I , I P Λ s be the ideal generated by the G IJ . Since K r T I ,I P Λ s J is the homogeneous coordinate ring of G p r, n q , then dim p K r T I , I P Λ s{ J q “ p r ` qp n ´ r q ` and Remark 2.4 yields dim p K r S, T I , I P Λ s{ J q “ p r ` qp n ´ r q ` “ dim p G p r, n q q ` rank p Pic p G p r, n q qqq “ dim p R p G p r, n q qq where we denote by J the ideal generated by the polynomials G IJ in K r S, T I , I P Λ s . We conclude that thereare no further relations in R p G p r, n q q and hence R p G p r, n q q “ K r S,T I ,I P Λ s J as claimed. (cid:3) Now, we are ready to compute the Mori chamber decomposition of Eff p G p r, n q q . Proposition 6.5.
Let G p r, n q be the blow-up of the Grassmannian G p r, n q at a point. Then we have that Eff p G p r, n q q “ x E, H ´ p r ` q E y , Nef p G p r, n q q “ x H, H ´ E y and Mov p G p r, n q q “ x H, H ´ rE y if n “ r ` x H, H ´ p r ` q E y if n ą r ` . Furthermore,
MCD p G p r, n q q and SBLD p G p r, n q q coincide and their walls are given by the divisors E, H, H ´ E, . . . , H ´ p r ` q E as represented in the following picture N MORI CHAMBER AND STABLE BASE LOCUS DECOMPOSITIONS 25
E H C ´ H ´ E C H ´ E C ... H ´ rE C r H ´ p r ` q E C ´ “ r E, H q , C “ Nef p G p r, n q q “ x H, H ´ E y , C i “ p H ´ iE, H ´ p i ` q E s for i “ , . . . , r ,where with the notation C i “ p H ´ iE, H ´ p i ` q E s we mean that the ray spanned by H ´ p i ` q E belongs to C i but the ray spanned by H ´ iE does not, and similarly with the notation C ´ “ r E, H q we mean that the rayspanned by E belongs to C ´ but the ray spanned by H does not.Proof. The claims on the effective, nef and movable cones follow from [MR18, Theorem 1.3]. Furthermore, by[MR18, Theorem 1.3] the decomposition displayed in the statement is the stable base locus decomposition of Eff p G p r, n q q . Now, by Proposition 6.3 all the generators of R p G p r, n q q appear in the walls of the stable baselocus decomposition of Eff p G p r, n q q , and then Proposition 5.2 yields that the Mori chamber and the stable baselocus decomposition of Eff p G p r, n q q coincide. (cid:3) References [ABCH13] D. Arcara, A. Bertram, I. Coskun, and J. Huizenga,
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Antonio Laface, Departamento de Matematica, Universidad de Concepción, Casilla 160-C, Concepción, Chile
E-mail address : [email protected] Alex Massarenti, Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 30, 44121Ferrara, ItalyInstituto de Matemática e Estatística, Universidade Federal Fluminense, Campus Gragoatá, Rua AlexandreMoura 8 - São Domingos, 24210-200 Niterói, Rio de Janeiro, Brazil
E-mail address : [email protected], [email protected] Rick Rischter, Universidade Federal de Itajubá (UNIFEI), Av. BPS 1303, Bairro Pinheirinho, 37500-903,Itajubá, Minas Gerais, Brazil
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